Further, which opponents you play against is not the only factor. When you play a certain board as NS, all NS players on this board are indirectly your opponents. If they have a better result than you have, your score goes down and vice versa. On the other hand all EW pairs are your allies. It is good for your score if they do well.
Now the aim of a good balance in a movement scheme is to take care that effectively you are compared the same number of times to all other pairs, or to put it better: with the same weights. This ideal is often not achievable. But we will see that in many cases existing movements may be changed by simple means, such that the balance is improved considerably.
If you are not interested in technical details, skip to the Results page. Just keep in mind we use a quantity Qf with values between 0 and 100 as a measure of the balance where Qf = 100 is perfect.
P = number of pairs (even) r = number of rounds t = P/2 = number of times a board is played b = number of boards per round h = t1 is half a top on a board N = P * (P1) /2 = number of pairs of pairs. S = t * r * b * h = "amount of competition" For simplicity we take b=1 (does not matter for the balance).Calculate, for each of the N possible combinations of pairs, a score which is the sum of contributions of each board, as follows:
We further call SS = the sum of the squares of all these scores. It is this number we want to minimize in order to optimize the balance.
The method explained above is taken from
https://www.ebu.co.uk/documents/media/bridgemovementsthemaths.pdf
https://chrisryall.net/memories/john.manning.htm
See also F.C. Schiereck, Groot Schemaboek N.B.B. Versie 2002

An modified version of the quality factor takes into account that the scores are integer values, so they cannot be exactly equal to S/N, since S/N is usually not an integer. We try to find a set of N integer numbers whose sum is S and and the sum of squares of which is minimal. This minimum of the sum of squares we use instead of S * S / N in the formula for Qc. We will use this modified version, called Qf, to be able to compare results with those of Schiereck. The difference between Qc and Qf is usually small.



So there is a 1 to 1 relationship between Q_{c} and s; s = 1 corresponds to Qc = 50. In words s is the spread of the scores divided by the average score.
One may represent the scores in a P by P matrix, the score matrix. It is a symmetrical matrix whose diagonal is undefined. Each element is the mutual score of two pairs.
Like Schiereck and Manning we use the word 'score' which may be confusing. A high 'score' for the combination pair1pair2 means a good result for pair1 is bad for pair2 and vice versa. 'Correlation' might be a better word.
Pair  Contract  Res.  Score  MP  
NS  EW  NS  EW  
1  3 NT  C  +600  3  
2   1   600  5  
3  3 NT  C  +600  3  
4   3   600  5  
5  3 NT  C  +600  3  
6   5   600  5  
7  3 NT  +1  +630  8  
8   7   630  0  
9  3 NT  C  +600  3  
10   9   600  5 
See the scorecard shown beside. The average MP score is h=4. Since pair 7 scores a top all other pairs in the same compass directions score 1 MP less, and pairs in the opposite direction 1 MP more (except of course pair 8 who score 0).
In other words, for the influence on your result, playing against a pair once is equivalent to playing h times in the same direction as this pair.

Meanwhile Gerrit van der Velde has been so kind as to put his program at my disposal (see Historical development). The strategy followed here is borrowed from a method used in solid state physics to model processes such as crystallization. Here the tables are switched in a random order. Not only profitable switches are accepted, but every now and then also switches that decrease the balance. When and how many depends on a "temperature" parameter. By alternatingly decreasing and increasing the temperature you hope to escape from secondary minima and finally to reach the absolute minimum. Again the overall best movement is preserved.
It proved that, especially for large numbers of tables, this strategy is much more effective. Therefore an adapted version of the main subroutine of this programme: has been adopted in "balans" as of version 4. A number of large movements from the tables "Multisession movements" and "Scrambled Mitchells" have been improved as a result.
Version 7 of balans is a major revision, and is a joint venture of Ulrik Dickow and me. In this version the optimalisation of the vacancy quality is introduced, and at the same time the strategy described above was extended and finetuned by Ulrik, resulting in a much faster program.
In version 7.4 he added possible user selection and tuning of the optimisation algorithm. The new optional slow cooling mode reverts to the simplest possible, classic 1983 simulated annealing. It is especially useful for difficult movements with some Howelllike content (pair A and B meet C but also each other), finding deeper minima than the default mode of fast temperature fluctuations with odd/even freezing, repeated retries and other tweaks aimed at producing reasonably good results very fast (impatient/frantic mode).
Is there any guarantee the absolute optimum is found? No there is not. Of all possibilities to switch n out of N tables only a limited number is investigated. In practice, for many small movements no improvement occurs after a few hundred iterations. But, certainly for large groups, there remains a chance of further improvements.
balans [options] <movement file>Some options are (for a complete list see file README_en.html):
c : check the movement, no optimalisation s n : do optimalisation for n iterations r n : keep row n fixed (= round n) t n : keep column n fixed (= table n) f n : keep the first n positions fixed h : show a short description of the optionsThe number of iterations is the number of trials for improvement. A few thousand is sufficient in most cases.
One often wants to keep the first round fixed. This may be done without
loss of generality, unless the number of tables is larger than the
number of board groups (more tables than rounds). In that case it is
still possible without problem to fix as many tables in round one as
there are board groups.
Option f is specially for this case.
Application of keeping a column fixed: in a movement of 14 pairs, 6 rounds, one pair (14 or 13) always sits at table 7. For this pair it is convenient if also the direction NS or EW is always the same. Thus you may try if this is feasible without disturbing the optimal balance.
The normal output of the program contains also the quality factors for the case of an absent pair, for all choices of which pair is absent.
8 6 6 6 0 1 2 A 3 4 B 5 6 C 7 8 D 0 0 0 0 0 0 3 6 A 1 7 B 0 0 0 0 0 0 5 8 E 4 2 F 7 5 A 8 6 B 1 4 C 2 3 D 0 0 0 0 0 0 4 8 A 0 0 0 0 0 0 1 6 D 2 7 E 5 3 F 0 0 0 5 2 B 3 7 C 0 0 0 6 4 E 1 8 F 0 0 0 0 0 0 8 2 C 5 4 D 1 3 E 7 6 FThe pairs should be numbered starting with 1, the characters to designate board groups may be chosen arbitrarily. In the example above the number of tables is larger than half the number of pairs. Therefore not all tables are in use simultaneously. Tables not in use are indicated by zeroes.
For an example of the output, click here.
The program in written in C, developed in the MinGW environment, and compiled with gcc.
Special thanks are due to Gerrit van der Velde for making his code available and to Joop van Wijk for critical comments and useful suggestions in all stages of this work.
There are two types of movements, (Short) Howells and Scheveningen. In both cases the arrangement in the first round is the same, the so called universal starting position. Hence, the scores of the first round may be kept without change, whenever you want to switch to another movement after this round, e.g. due to absentees.
In the Howell movements pair 1 is always stationary, NS at table 1. You might want to use this for a person who has problems with walking or who needs special provisions. For the rest, players and boards move after every round. The number of tables is minimal. For 10 pairs you need only 5 tables, not 6 or 7.
In the Scheveningen movements all boards always remain on their original table, except if they have to be shared. In a movement of 7 rounds and 10 pairs therefore 7 tables are used. At appendix tables there always is a stationary pair.
We have paid attention to the requirement that a good balance should remain when one pair is absent.
In the following table Qf is the quality factor for an even number of pairs, Qf_{1} the quality factor if the highest pair number is absent.
T E A M M O V E M E N T S
movement Qf Qf1 Howell Sch pairs rounds 6 6 83.33 80 y 8 6 80 71.74 y y See 8p6rGSBopt.asc for a better choice 10 6 83.76 75.79 y y 12 6 79.75 71.81 y y 14 6 70.41 70.36  y 16 6 76.92 71.92  y 8 7 100 100 y y 10 7 84.94 82.93 y y 12 7 76.87 75.82 y y 14 7 92 85.09 y y 16 7 78.74 72.17  y 18 7 63.95 62.38  y 8 8 87.50 86.36 y  10 8 89.51 82.50 y y 12 8 85.38 78.33 y y 14 8 82.46 78.15 y y 16 8 94.64 88.64 y y
A note of warning: only use this type of movement for events with a fixed group of participants. As a result of the optimization over multiple session the balance per session deteriorates, such that there is no point in using them for a variable number of participants.
pairs rounds sessions rounds per session encounters Qf 10 18 2 9 2 100 100 monster 10 18 3 6 2 100 100 monster 12 11 2 5 en 6 1 100 100 gedrocht 12 11 3 3 en 4 1 100 100 gedrocht 12 24 4 6 2 à 3 99.24 96.97 vals monster (nov. 2008) 12 30 5 6 2 à 3 99.55 97.40 vals monster (nov. 2008) 14 26 2 13 2 100 100 monster 14 26 4 6 and 7 2 99.03 100 gedrocht (nov. 2008) 14 24 4 6 1 à 2 98.74 96.32 vals monster (nov. 2008) 14 30 5 6 2 à 3 99.26 96.15 vals monster (nov. 2008) 14 36 6 6 2 à 3 99.52 97.74 vals monster (nov. 2008) 14 39 6 6 and 7 3 99.38 100 gedrocht 16 15 2 7 and 8 1 100 100 gedrocht 16 15 3 5 1 100 100 monster (Sep. 2007) 16 30 5 6 2 99.12 100 monster (Sep. 2007) 18 17 3 6, 6, 5 1 94.74 100 gedrocht 20 19 3 6, 6, 7 1 96.10 100 gedrocht 22 21 3 7 1 95.98 100 monster 24 23 3 8, 8, 7 1 96.58 100 gedrocht some of the above also appear in the section on "perfect movements" below.
The movements listed above are avialable on the downloadpage, click on "monster_pakket.exe". Again the uploading into Dutch scoringprograms may be turned off by unchecking "toevoegen aan NBB_R".
In a Mitchell movement the field may be separated into NS and EW, each with their own winner. Besides the Standard Mitchell other types of movements with this property will be considered in this section. Such movements may be modified by "arrow switches" so as to obtain a balanced movement for a contest with one winner. We will limit ourselves to "complete" movements, where the number of rounds is equal to the number of tables, and all pairs play all boards.
A simple variation on this theme is:
 The NS pairs go up one table
 The EW pairs go down one table
 The boards stay where they are.
Schemes like this work fine when the number of tables is odd. But for an even number of tables duplications occur after the first half the session. An effective solution to this problem is the Relay Mitchell. A disadvantage is that board sharing occurs in each round.
A better solution is the Double Weave Mitchell. The movement is more complicated but board sharing is avoided. However, this method only works when the number of tables is a multiple of 4, i.e. 4, 8, 12 ...
Another option for an even number of tables is the Skip Mitchell. Here the moving pairs do not meet all the stationary pairs. This inferior movement will not be considered further.
For a description of the above mentioned movements see "Movements  a fair approach" by Hallén, Hanner en Jannersten.
We will also look at some "Scheveningen" movements, taken from "Groot Schemaboek 2002" by F.C. Schiereck. Similar to the Mitchell, each NS pair meets only EW pairs as opponents. The boards are stationary, the NS pairs are labelled with odd numbers, and EW have even numbers. For an odd number of tables these movements are equivalent to the Standard Mitchell. For an even number board sharing is avoided in most cases as opposed to the Relay Mitchell.
1 8 A 2 9 B 310 C 411 D 512 E 613 F 714 G 114 B 2 8 C 3 9 D 410 E 511 F 612 G 713 A 113 C 214 D 3 8 E 4 9 F 510 G 611 A 712 B 112 D 213 E 314 F 4 8 G 5 9 A 610 B 711 C 111 E 212 F 313 G 414 A 5 8 B 6 9 C 710 D 110 F 211 G 312 A 413 B 514 C 6 8 D 7 9 E 1 9 G 210 A 311 B 412 C 513 D 614 E 7 8 FA pleasure to the eye and also perfectly balanced, when you calculate a result for NS and EW separately. This is evident from the score matrix.
\ 7 7 7 7 7 7 0 0 0 0 0 0 0 7 \ 7 7 7 7 7 0 0 0 0 0 0 0 7 7 \ 7 7 7 7 0 0 0 0 0 0 0 7 7 7 \ 7 7 7 0 0 0 0 0 0 0 7 7 7 7 \ 7 7 0 0 0 0 0 0 0 7 7 7 7 7 \ 7 0 0 0 0 0 0 0 7 7 7 7 7 7 \ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \ 7 7 7 7 7 7 0 0 0 0 0 0 0 7 \ 7 7 7 7 7 0 0 0 0 0 0 0 7 7 \ 7 7 7 7 0 0 0 0 0 0 0 7 7 7 \ 7 7 7 0 0 0 0 0 0 0 7 7 7 7 \ 7 7 0 0 0 0 0 0 0 7 7 7 7 7 \ 7 0 0 0 0 0 0 0 7 7 7 7 7 7 \If you only want one winner this is not satisfactory. The problem may be solved by switching NS and EW at all tables during one round, a socalled arrow switch round. It does not matter which round is switched. For instance:
1 8 A 2 9 B 310 C 411 D 512 E 613 F 714 G 14 1 B 8 2 C 9 3 D 10 4 E 11 5 F 12 6 G 13 7 A 113 C 214 D 3 8 E 4 9 F 510 G 611 A 712 B 112 D 213 E 314 F 4 8 G 5 9 A 610 B 711 C 111 E 212 F 313 G 414 A 5 8 B 6 9 C 710 D 110 F 211 G 312 A 413 B 514 C 6 8 D 7 9 E 1 9 G 210 A 311 B 412 C 513 D 614 E 7 8 FBy this simple measure the quality factor Qf shoots up from 46.94 to no less than 92. The new score matrix becomes:
\ 3 3 3 3 3 3 4 4 4 4 4 4 0 3 \ 3 3 3 3 3 0 4 4 4 4 4 4 3 3 \ 3 3 3 3 4 0 4 4 4 4 4 3 3 3 \ 3 3 3 4 4 0 4 4 4 4 3 3 3 3 \ 3 3 4 4 4 0 4 4 4 3 3 3 3 3 \ 3 4 4 4 4 0 4 4 3 3 3 3 3 3 \ 4 4 4 4 4 0 4 4 0 4 4 4 4 4 \ 3 3 3 3 3 3 4 4 0 4 4 4 4 3 \ 3 3 3 3 3 4 4 4 0 4 4 4 3 3 \ 3 3 3 3 4 4 4 4 0 4 4 3 3 3 \ 3 3 3 4 4 4 4 4 0 4 3 3 3 3 \ 3 3 4 4 4 4 4 4 0 3 3 3 3 3 \ 3 0 4 4 4 4 4 4 3 3 3 3 3 3 \Do not make the mistake of switching two rounds. Then the balance collapses and Qf goes down to 49.64
How this case works out in imaginary contests with one, and also with two very strong pairs we will see further on.
For an even number of tables such a regular scheme is not possible and tricks are necessary to avoid that pairs meet the same opponents or the same boards twice. For 6 tables the Relay Mitchell looks as follows:
1 7 A 2 8 B 3 9 C 410 E 511 F 612 A 112 B 2 7 C 3 8 D 4 9 F 510 A 611 B 111 C 212 D 3 7 E 4 8 A 5 9 B 610 C 110 D 211 E 312 F 4 7 B 5 8 C 6 9 D 1 9 E 210 F 311 A 412 C 5 7 D 6 8 E 1 8 F 2 9 A 310 B 411 D 512 E 6 7 Fwhere table 1 and 6 share boards in all rounds. An alternative is the following movement where board sharing is limited to the last two rounds.
1 2 A 3 4 B 5 6 C 7 8 D 910 E 1112 F 11 6 A 712 B 110 C 9 4 D 5 8 E 3 2 F 3 8 A 9 6 B 7 4 C 11 2 D 112 E 510 F 5 4 A 1110 B 312 C 1 6 D 7 2 E 9 8 F 512 D 1 8 B 9 2 C 310 D 11 4 E 7 6 F 912 A 5 2 B 11 8 C 710 A 3 6 E 1 4 FThis is the Scheveningen12 on pag 5. of the Groot Schemaboek.
We see that one single arrow switch round is optimal for complete Mitchells of 12 and 14 pairs. But this is not a general rule. For a Mitchell for 22 pairs and 11 rounds, for instance, one arrow switch (Qf=89.2) is still better than 2 (Qf=85.04), but further optimalisation leads to Qf=93.41.
In the following table we present Quality factors for Mitchells and Mitchelllike movements that are "scrambled" by arrow switches.
revised April 2017 Nr of rounds Type nr of arrow switch rounds best value = tables 0 1 2 3 5 M 46.00 67.65 31.08 (2 5) 74.68 *) 6 RM 46.67 79.75 42.86 (2 3) 84.00 **) 6 GSB 46.67 84.00 40.65 (2 6) 84.00 7 M 46.94 92.00 49.64 (2 7) 92.00 8 RM 47.32 91.38 60.92 (2 7) 91.38 8 DWM 47.32 94.64 57.61 (2 5) 94.64 ***) 8 GSB 47.32 94.64 55.21 (2 8) 94.64 9 M 47.53 93.90 70.00 (2 7) 93.90 10 RM 47.78 90.03 78.93 (2 10) 93.80 10 GSB 47.78 91.88 77.12 (2 10) 93.80 11 M 47.93 89.23 85.04 (2 8) 93.41 12 RM 48.11 85.55 89.58 (2 12) 67.08 (2 3 12) 94.00 12 DWM 48.11 86.59 89.86 (2 4) 65.24 (2 4 8) 94.00 12 GSB 48.11 86.59 89.86 (2 12) 65.24 (2 3 12) 94.00 13 M 48.22 84.02 92.97 (2 8) 72.99 (2 7 8) 93.96 14 RM 48.35 81.08 94.27 (2 5) 79.06 (2 5 6) 94.77 14 GSB 48.35 81.70 94.44 (2 8) 77.36 (2 8 11) 94.77 15 M 48.44 79.56 95.49 (2 9) 83.03 (2 8 9) 95.49 notes. M = Standard Mitchell RM = Relay Mitchell DWM = Double Weave Mitchell GSB = "Scheveningen" movement from "Groot Schemaboek 2002" by F.C. Schiereck The numbers in between brackets denote the switch rounds used in the calculation. *)The optimum balance for the 5table Mitchell is obtained by switching one round, for 3 or 4 tables. **)The optimum balance for the 6table Mitchell is obtained by switching one round, for all tables except either table 1 or 6 (the sharing tables). ***)The optimum Qf for the 8table DWM is obtained by arrow switching any round, but only rounds 1,4,5,8 give also optimum vacancy quality.
After all, the fact that one arrow switch is usually necessary and sufficient turned out to be nothing new.
The example of the 7table Mitchell we started this paragraph with is borrowed from John Manning, who remarks about the result: "The standard deviation works out at 1.05 and cannot be further reduced by switching more or fewer boards".
Already in 1979 John Manning paid attention to the problem of the optimum number of arrow switches and indicated "A rough and ready rule is to switch about one eighth of the boards in a Mitchelltype movement."
John Probst investigated this problem mathematically and generally and comes to the same conclusion:
We must arrow switch slightly more than 1/8 of the rounds for fairness. Anything else is WRONG!!!
see:
J.Manning:
The Mathematics of Duplicate Bridge Tournaments
(Bulletin of Institute of Mathematics and its Applications Vol. 15, No. 8/9, August/September 1979, pp201  206)
J.Probst: https://www.blakjak.org/why_1in8.htm
See also: https://www.blakjak.org/lws_men1.htm
Ross Moore:
"Too many arrowswitches spoil the balance"
(1992) comes to the same conclusion.
But if the following Mitchell movement was used:
1 7 A 2 8 B 3 9 C 410 D 511 E 612 F 111 C 2 7 F 310 E 412 B 5 8 D 6 9 A 1 9 D 212 C 3 8 A 4 7 E 510 F 611 B 112 E 210 A 311 F 4 8 C 5 9 B 6 7 D 1 8 F 2 9 E 3 7 B 411 A 512 A 610 C 110 B 211 D 312 D 4 9 F 5 7 C 6 8 Ethe result looks as follows:
pair 1 100.00 pair 2 40.00 pair 3 40.00 pair 4 40.00 pair 5 40.00 pair 6 40.00 pair 7 50.00 pair 8 50.00 pair 9 50.00 pair 10 50.00 pair 11 50.00 pair 12 50.00As we see the other pairs score either 40 or 50%. The pairs that were beaten by pair 1 still get 50% while those who did not even meet this pair get only 40%.
We now improve the movement as follows. In round 3 we do an arrow switch except at table 1.
1 7 A 2 8 B 3 9 C 410 D 511 E 612 F 111 C 2 7 F 310 E 412 B 5 8 D 6 9 A 1 9 D 12 2 C 8 3 A 7 4 E 10 5 F 11 6 B 112 E 210 A 311 F 4 8 C 5 9 B 6 7 D 1 8 F 2 9 E 3 7 B 411 A 512 A 610 C 110 B 211 D 312 D 4 9 F 5 7 C 6 8 EThen the result becomes:
pair 1 100.00 pair 2 43.33 pair 3 43.33 pair 4 43.33 pair 5 43.33 pair 6 43.33 pair 7 46.67 pair 8 46.67 pair 9 50.00 pair 10 46.67 pair 11 46.67 pair 12 46.67Pair 1 plays exactly the same boards against the same opponents as in the original movement. But now the results are much closer together. The deviations from the ideal value 45.45 are roughly twice as small. These smaller fluctuations are the result of the improved balance. The original movement has quality factor Qf=46.67, standard deviation sd=2.99, the second one Qf=84, sd=1.29.
We next show the disastrous effect of too much switching:
movement with 2 arrow switches:
1 7 A 2 8 B 3 9 C 410 D 511 E 612 F 111 C 2 7 F 310 E 412 B 5 8 D 6 9 A 9 1 D 12 2 C 8 3 A 7 4 E 10 5 F 11 6 B 12 1 E 10 2 A 11 3 F 8 4 C 9 5 B 7 6 D 1 8 F 2 9 E 3 7 B 411 A 512 A 610 C 110 B 211 D 312 D 4 9 F 5 7 C 6 8 EAgain we assume pair 1 always scores a top and on all other tables an equal average result is obtained. With this movement we get the result:
pair 1 100.00 pair 2 53.33 pair 3 53.33 pair 4 46.67 pair 5 53.33 pair 6 46.67 pair 7 50.00 pair 8 36.67 pair 9 43.33 pair 10 36.67 pair 11 36.67 pair 12 43.33We can not eliminate all accidental factors in duplicate bridge. When your opponents are the only ones to bid and make a cold slam there is nothing you could have done about it, yet you get a 0. But accidental factors as a result of a poor balance may be reduced considerably by using suitable movement schemes.
Tips:
arrow switches none 1 2 3 Qf 46.94 92 49.64 32.71 pair 1 100.00 100.00 100.00 100.00 pair 2 41.67 46.43 46.43 46.43 pair 3 41.67 46.43 51.19 51.19 pair 4 41.67 46.43 51.19 55.95 pair 5 41.67 46.43 51.19 55.95 pair 6 41.67 46.43 51.19 51.19 pair 7 41.67 46.43 46.43 46.43 pair 8 50.00 45.24 45.24 40.48 pair 9 50.00 45.24 40.48 40.48 pair 10 50.00 45.24 40.48 40.48 pair 11 50.00 45.24 40.48 40.48 pair 12 50.00 45.24 45.24 45.24 pair 13 50.00 45.24 45.24 40.48 pair 14 50.00 50.00 45.24 45.24Again, we see in column 1 the perfect balance, as long as we work out separate scores for the two groups NS and EW. But for a joint result this is of course no good. In column 2, one arrow switch, the results of pairs who played against pair 1, and those that played in the same direction, are nicely and closely together with one single exception. In columns 2 and especially 3 we see again the effect of overcompensation. For completeness' sake: the arrow switches in the above are in rounds 2, 2+3, 2+3+4, respectively.
In some cases it is even possible to split the movement into two or more sessions, each with their own board sets, and still maintain perfect balance.
For an odd number of tables a perfect balance may be sometimes be obtained when the number of board sets is doubled. An example is the Howell6 with Qf=100 given in "Movements a Fair Approach". This movement has Qf1av= 66.67, Qf1max= 100. A considerable improvement is found by optimizing the "vacancy quality". With program "balans" one finds in no time a superperfect movement with Qf1av=100. However now one has an arrow switch in midround in a few cases. In this superperfect variant this occurs for everyone, in only one round.
A survey of some goodquality movements is given below.
4 pairs 3 rounds 1 session Qf=100 4 pairs 6 rounds 1 session Qf=100 same pairs meet twice 6 pairs 5 rounds 1 session Qf=100 2 board groups per round 8 pairs 7 rounds 1 session Qf=100 or this alternative 10 pairs 9 rounds 1 session Qf=100 2 board groups per round 10 pairs 18 rounds 2 sessions Qf=100 10 pairs 18 rounds 3 sessions Qf=100 12 pairs 11 rounds 1 session Qf=100 12 pairs 11 rounds 2 sessions Qf=100 12 pairs 11 rounds 3 sessions Qf=100 14 pairs 13 rounds 1 session Qf=100 2 board groups per round 14 pairs 26 rounds 2 sessions Qf=100 16 pairs 15 rounds 1 session Qf=100 16 pairs 15 rounds 2 sessions Qf=100Some of these movements, in particular the Howell10's are due to Joop van Wijk.
The concept of "balance" probably is almost as old as competitive bridge. De mathematical foundation of the work on balance and arrow switches presented here was laid by John Manning (1979) in an article with the title The Mathematics of Duplicate Bridge Tournaments.
In the book : "Movements  a fair approach" (1994) one finds discussions that might have lead to same the model outlined above, but as noticed, in a practical application the authors miss the right track. Apparently the work of Manning was unknown to the authors.
Paul Vermaseren developed, in the 80's, an effective computer programme for the optimalisation of Howell schemes. The results did not get the attention they deserved. A few perfectly balanced "Howells in several sessions" appeared in Wekowijzer. The Howell for 16 pairs in 3 sessions given above is one of those results.
The "least squares" approach used here I encountered in the Groot Schema Boek by Schiereck (2002) and on John Manning's web page. Both of these developments apparently took place independently. The person chiefly responsible for the results in the Groot Schema Boek is Gerrit van der Velde, who already in the 90's developed a computer programme to optimize the balance. Parts of this programme are used in "balans".
The "1 in 8" rule for the optimal number of arrow switches in Mitchell Movements is found already in the article of Manning (1979) cited above. It was further propagated by the work of John Probst on this subject. The least squares approach to characterize the balance, and recommendations about the optimal number of arrow switches were also given by Ross Moore (1992).
However there seem to be many places on earth where these results are still largely unknown.
The idea of "vacancy quality", and its use for a further improvement, rose in discussions with Ulrik Dickow and is introduced for the first time in the current work.